Description: Subgroups are an algebraic closure system. (Contributed by Stefan O'Rear, 4-Apr-2015) (Revised by Mario Carneiro, 22-Aug-2015)
Ref | Expression | ||
---|---|---|---|
Hypothesis | subgacs.b | |
|
Assertion | subgacs | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subgacs.b | |
|
2 | eqid | |
|
3 | 2 | issubg3 | |
4 | 1 | submss | |
5 | 4 | adantl | |
6 | velpw | |
|
7 | 5 6 | sylibr | |
8 | eleq2w | |
|
9 | 8 | raleqbi1dv | |
10 | 9 | elrab3 | |
11 | 7 10 | syl | |
12 | 11 | pm5.32da | |
13 | 3 12 | bitr4d | |
14 | elin | |
|
15 | 13 14 | bitr4di | |
16 | 15 | eqrdv | |
17 | 1 | fvexi | |
18 | mreacs | |
|
19 | 17 18 | mp1i | |
20 | grpmnd | |
|
21 | 1 | submacs | |
22 | 20 21 | syl | |
23 | 1 2 | grpinvcl | |
24 | 23 | ralrimiva | |
25 | acsfn1 | |
|
26 | 17 24 25 | sylancr | |
27 | mreincl | |
|
28 | 19 22 26 27 | syl3anc | |
29 | 16 28 | eqeltrd | |